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While introducing myself to differential equations, I read that the solution to a differential equation may contain an arbitrary constant without being a general solution.

I have been solving initial value problems under the concept of anti-differentiation for a long time now. I am a little aware of general and particular solutions from my engineering classes. I have always thought that a solution having an arbitrary constant cannot be anything other than a general solution. I don't think a particular solution can have an arbitrary constant in it. The lines that I read today left me curious. I am really curious to see these equations and maybe know how they work.

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  • $\begingroup$ If you face a differential equation of order $n$, the general solution contains $n$ constants. Now, if you are given $m$ conditions $(m \leq n)$, $m$ of these constants "disappear" since fixed by the conditions. $\endgroup$
    – Claude Leibovici
    Sep 10, 2017 at 6:55
  • $\begingroup$ So, the solution, having some constants, is now a particular solution under a few conditions( m ), right? $\endgroup$
    – R004
    Sep 10, 2017 at 6:58
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    $\begingroup$ I think what your text is trying to say is that a solution containing the right number of arbitrary constants, although it is a "general" solution (not a particular solution), yet it may fail to be the most general solution; i.e., there may be particular solutions which can not be obtained by assigning values to the arbitrary constants. $\endgroup$
    – bof
    Sep 10, 2017 at 7:01
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    $\begingroup$ For example, the (non-linear) first order equation $$\frac{dy}{dt}=y^2$$ has the "general solution" $$y=\frac1{C-t}$$ but it also has the "singular solution" $$y=0$$ which is not covered by the "general solution". $\endgroup$
    – bof
    Sep 10, 2017 at 7:05
  • $\begingroup$ This is a very interesting example. I appreciate it. It does justify your "most general solution" statement. But there still is a room for doubt about whether the text is exactly conveying what you just did. $\endgroup$
    – R004
    Sep 10, 2017 at 7:13

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One can find examples (not exclusively) when the solution includes multi-valuated functions, which is the sometimes the case of inverse functions. For example, the ODE : $$\cos(y(x))\frac{dy}{dx}=1$$ has this family of solutions : $$y(x)=\sin^{-1}(x+c_1)$$ This solution includes an arbitrary constant $c_1$ but is not the general solution which is : $$y(x)=\sin^{-1}(x+c_1)+2\pi\:n$$

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  • $\begingroup$ This is a great example! $\endgroup$
    – R004
    Sep 10, 2017 at 9:45
  • $\begingroup$ Lastly, the text mentioned that establishing that a solution "is" general requires deeper knowledge of the theory of differential equations. Although I am a beginner at differential equations, could I, if you could provide, take a peak at the procedure? $\endgroup$
    – R004
    Sep 10, 2017 at 9:49
  • $\begingroup$ I am afraid that it is not possible to answer on a general manner. There is no general method to solve all kind of differential equations and so, for the generality of the solutions found in all cases. But they are some particular methods that you will progressively learn. $\endgroup$
    – JJacquelin
    Sep 10, 2017 at 11:00

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